1. Field of the Invention
The present invention relates to sigma-delta modulation.
2. Description of the Related Art
A Sigma-Delta-Modulator (SDM) transforms a bandlimited input signal into a digital output signal. The input signal modulates the output pulse density. FIG. 1 illustrates an example of a SDM signal. The input signal can be reclaimed by low-pass filtering the output signal. FIG. 2 illustrates the basic structure of a conventional SDM 10. The SDM 10 includes an adder 12, a loop filter 14, and a quantizer 16. SDMs may be implemented as analog or digital SDMs.
The clock frequency fa of the SDM 10 should be much higher than the highest frequency of the input signal Ue. To get a sufficiently high signal-to-noise-ratio from the digital output of a CD-player (16 bit/44.1 kHz) oversampling by a factor of at least 32 is generally necessary. For a digital SDM, the input signal should be supplied with this high data rate. This may be achieved by using a digital interpolation filter.
In the simplest case, the loop filter 14 may be implemented as an integrator. FIGS. 3 and 4 illustrate an analog 141 and a digital 142 integrator, respectively.
The loop filter 14 determines the resolution (S/N-ratio) of the SDM 10. By using a loop filter 14 of higher order, a better S/N-ratio can be achieved, but stability problems may arise.
The threshold of the quantizer 16 is normally zero. The output signal Ua of the quantizer 16 is +1, if the input signal to the quantizer 16 Uk>0 and −1, if Uk<0. The quantizer 16 changes its output with each new clock cycle.
As an example, an analog SDM with an input signal of zero and a first order loop filter operates as follows. When starting, the SDM 10 output is +1. The loop back to the input provides a new input to the loop filter 14 of −1 (input signal (0)−output signal (+1)=−1). The output of the loop filter 14 Uk slowly decreases to the negative supply rail. Therefore, the next clock cycle sets the quantizer 16 output to −1. This gives a new input to the loop filter 14 of +1 (input (0)−output (−1)=+1). Uk now drifts to positive values. The output of the SDM 10 now is a random stream of bits (+1 and −1). FIG. 5 illustrates the values of Ua and Uk at various times. If the input signal Ue of the SDM 10 is zero, the average of the output Ua is also zero. When modulating the SDM 10, the appearance and sequence of +1 and −1 pulses changes accordingly to the input signal Ue.
To examine the SDM 10 in the frequency domain, it advantageous to substitute the quantizer 16 with an adder 18, a noise source N(z), and a quantizing amplifier 20 with gain gQ. From FIG. 6 two transfer functions are obtained, the signal transfer function HX(z) and the noise transfer function HN(z).If                     z        =                  ⅇ                      j            ⁢                                                   ⁢                          ω                              ω                g                                                                        (        1        )            then:                                           H            X                    ⁡                      (            z            )                          =                                            Y              ⁡                              (                z                )                                                    X              ⁡                              (                z                )                                              =                                                    g                Q                            ⁢                                                H                  lf                                ⁡                                  (                  z                  )                                                                    1              +                                                g                  Q                                ⁢                                                      H                    lf                                    ⁡                                      (                    z                    )                                                                                                          (        2        )                                                      H            N                    ⁡                      (            z            )                          =                                            Y              ⁡                              (                z                )                                                    N              ⁡                              (                z                )                                              =                      1                          1              +                                                g                  Q                                ⁡                                  (                                                            H                      lf                                        ⁡                                          (                      z                      )                                                        )                                                                                        (        3        )            where gQ is the quantizer gain.
FIG. 7 illustrates two different noise transfer functions HN(z) for a SDM 10 with a 4th order loop filter with different gains. The noise transfer function HN(z) shows a strong rejection of the frequencies in the audio range (f/fa=0 . . . 0.01). High frequencies are amplified and passed through. This is also called noise-shaping. The noise level in the audio frequency band may be lowered by increasing the order of the loop filter 14.
There are three possible implementations of a SDM 10, an analog SDM, a digital SDM, and an SDM with a switched capacitor filter (SC-filter). Different applications require different implementations. For example, for an A/D-converter, either the analog SDM or the SC-SDM would be appropriate. For a D/A-converter, the digital SDM is the best choice.
Conventional SDMs are well-known for their insensitivity to analog imperfections, and therefore they are appropriate for a large number of applications. Their usefulness has led to the adoption of the Direct Stream Digital, or DSD, format (the single bit output of an SDM) as the data format for Super Audio CD (SACD). It is believed that the data stream on the SACD can be derived from the A/D conversion step with minimal additional signal processing, thus leading to the highest quality possible. However, conventional SDMs themselves produces signal artifacts, which are due to the non-linear character of the quantizer 16 in the SDM 10. These effects are shown in FIG. 8, where odd harmonic distortion products can be clearly observed.